Abstract
We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.
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Kruml, D., Pelletier, J.W., Resende, P. et al. On Quantales and Spectra of C*-Algebras. Applied Categorical Structures 11, 543–560 (2003). https://doi.org/10.1023/A:1026106305210
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DOI: https://doi.org/10.1023/A:1026106305210